Breakthrough Tracker record
A claimed five-variable counterexample to the Hessian conjecture
The preprint gives an explicit degree-14 integer polynomial in five variables with constant Hessian determinant 128 and two distinct rational points having the same gradient. If the identities and bridge arguments stand, the Hessian conjecture is false in every dimension at least five.
← Back to the filtered Breakthrough Tracker
- Stable ID
math-hessian-conjecture-five-variable-counterexample-2026- Revision
math-hessian-conjecture-five-variable-counterexample-2026.v1- Field
- Mathematics · Algebraic geometry
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed counterexample
- Last checked
AI role
The authors say Liang Yang found the Schur-descent step with ChatGPT Pro assistance and that exact-arithmetic checks used Claude. The authors wrote the paper and accept responsibility for its claims.
Record details
- Problem or result
- The Hessian conjecture in dimension five and its stabilization to higher dimensions
- Authors
- Guowu Meng and Liang Yang
- Institutions
- Hong Kong University of Science and Technology; Sichuan University
- Result date
- Version one dated July 23 and submitted July 24, 2026
Why it matters
Together with known results in dimensions one through three, the construction would leave only the four-variable Hessian conjecture open; that case is linked to the still-open two-variable Jacobian conjecture.
Limits
The draft appeared only days after the Jacobian counterexample on which it builds. It is not peer reviewed, and the authors explicitly request independent computer-algebra checking and expert review. The four-variable Hessian and two-variable Jacobian cases remain open.
Sources
Correction and revision history
- 2026-07-27 — Added as a provisional claimed counterexample with the explicit polynomial test, AI contribution, unsettled low-dimensional cases and authors’ call for independent checking preserved.
Machine-readable: JSON v1 · CSV v1 · Schema v1
This individual record remains noindex until a story-specific featured image passes Kingy’s rendered-pixel visual review. The source-linked tracker hub remains the public index.